Isotropic Basis in a Real Symmetric Bilinear Space: Existence and Construction - Biswas, Debapriya; Das, Himadri Lal (Armenian Journal of Mathematics, 2025)
A basis of a real symmetric bilinear space is called an isotropic basis if all its elements are isotropic. In this paper, we provide both necessary and sufficient conditions for the existence of such an isotropic basis. We present one geometric method and two linear algebraic methods for constructing isotropic bases. Additionally, we address a question  arising from the properties of symmetric bilinear forms. As a consequence, we explore various properties of the vector space spanned by the preimage set of a point under a real-valued continuous function. We also demonstrate some applications of these properties within the context of real symmetric bilinear spaces.
Keywords: Symmetric Bilinear Form; Symmetric Bilinear Space; Isotropic Basis; Preimage Dimension